Ad infinitum is a Latin

Latin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 phrase meaning "to infinity

Infinity

Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

."

In context, it usually means "continue forever, without limit" and thus can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses. It may also be used in a manner similar to the Latin phrase "et cetera

Et cetera

Et cetera is a Latin expression that means "and other things", or "and so forth". It is taken directly from the Latin expression which literally means "and the rest " and is a loan-translation of the Greek "καὶ τὰ ἕτερα"...

" to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:

• "The sequence 1, 2, 3, ... continues ad infinitum."
• "The perimeter of a fractal
  Fractal
  A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
  
   may be iteratively drawn ad infinitum."
• The 17th century writer Jonathan Swift
  Jonathan Swift
  Jonathan Swift was an Irish satirist, essayist, political pamphleteer , poet and cleric who became Dean of St...
  
   mocked the idea of self-similarity
  Self-similarity
  In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
  
   in natural philosophy with the following lines in his poem 'On Poetry: A Rhapsody':

• The Victorian era
  Victorian era
  The Victorian era of British history was the period of Queen Victoria's reign from 20 June 1837 until her death on 22 January 1901. It was a long period of peace, prosperity, refined sensibilities and national self-confidence...
  
   mathematician Augustus De Morgan
  Augustus De Morgan
  Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
  
   expanded on this with a similar verse

This text appears in the nursery rhyme The Siphonaptera

The Siphonaptera

"The Siphonaptera" is a nursery rhyme, sometimes referred to as Fleas.Big fleas have little fleas,Upon their backs to bite 'em,And little fleas have lesser fleas,and so, ad infinitum.Sometimes a second verse appears, with lines such as...

 almost unchanged.

See also

• The song that never ends
  The Song That Never Ends
  "The Song That Never Ends" is a self-referential and infinitely iterative children's song. The song is a single verse long, written in an infinite-loop motif in a march style, such that it naturally flows in a cyclical fashion, repeating the same verse over and over. It is very popular with...
  
  
• Turtles all the way down
  Turtles all the way down
  "Turtles all the way down" is a jocular expression of the infinite regress problem in cosmology posed by the "unmoved mover" paradox. The phrase was popularized by Stephen Hawking in 1988. The "turtle" metaphor in the anecdote represents a popular notion of a "primitive cosmological myth", viz...
  
  
• Recursion
  Recursion
  Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
  
  
• Self-reference
  Self-reference
  Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
  
  
• Mathematical induction
  Mathematical induction
  Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...